A conventional non-threaded bicycle headset assembly generally includes a steering tube which has smooth outer surface and extends through a head tube of the bicycle. The headset assembly is connected on each of two ends of the head tube and the steering tube rotatably extends through the headset assembly such that the steering tube can be rotated relative to the head tube by the headset assembly. A pressing member is mounted to a top end of the steering tube and an urging unit is forced fitting in the steering tube. A bolt is connected to the urging unit and a head of the bolt is engaged with a top end of the pressing member such that when rotating the bolt, the urging unit expands outward to securely connect the steering tube to the pressing member which is then moved downward to press on the headset assembly to position the bearing in the headset assembly.
Nevertheless, a special tool is required to hammer the urging unit into the steering tube and this takes a lot of time. Frequently, the urging unit is not forced into the steering tube as expected and this could affect the positioning of the headset assembly. Once the urging unit is deformed  because of the hammering, it is difficult to take it out and this affects the time required when the headset assembly needs to be replaced with a new one.
The present invention intends to provide a simple structure of the headset assembly which is easily to be assembled to the head tube and the steering tube.